AP Calculus BC glossary

Arc length

Arc length is the distance measured along a curve rather than straight across. It comes from adding up infinitesimal hypotenuses, which produces an integral of the square root of one plus the square of the derivative.

L = \int_a^b \sqrt{1 + \left(f'(x)\right)^2}\,dx

Each tiny piece of curve is nearly a straight segment with horizontal leg $dx$ and vertical leg $dy$ , so its length is $\sqrt{dx^2 + dy^2}$ . Factoring out $dx$ gives the formula.

In parametric form the same reasoning gives $L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt$ , which is also the integral of speed and therefore the total distance travelled.

Usually calculator work

Arc length integrands rarely have elementary antiderivatives, so these almost always appear on the calculator-active section.

Appears in: Unit 8: Applications of Integration, Unit 9: Parametric, Polar, and Vector (BC)